I think Reddit moderators are less restricted/directed by Reddit so they fall closer on the spectrum to the guy running a debate club/whatever at the local community centre. I don't think anyone would argue that they're an employee of the community centre, even if the community centre prohibited alcohol at their events.
I like your point, but I go back and forth on this. Rules are different for non-profits and volunteers, so let's make sure we're focusing on a for-profit function hall, not a local community center.
As far as I can tell, Reddit exerts minimal control over its moderators, and seems to treat them like normal users who happened to get more buttons to press (perhaps by design for this very issue). That seems to me like they're not unpaid employees.
At the same time, I'm reasonably certain that if the moderators of, say the "aww" subreddit, decided as a whole that they were only going to use their moderator powers 5 times a day each, Reddit would step in.
And that brings me back around the other way. Reddit relies on its moderators, for sure.
But is the lack of control they exert over mods the true status quo? Or is the lack of control just because it's convenient?
If I'm running debate club at the function hall and it gets wildly popular, the biggest scene around. I'm sure the owners would be thrilled. If I then decide that anyone who says "um" more once is no longer welcome, I imagine the popularity would wane. The hall might be really cranky about lost concessions. They might try to find someone else to run their own debate club and counter-program mine. They might even cancel my lease and get a new debate club going in the same room. But they're not going to step in and replace me as the moderator of my own debate club.
Nobody would argue that they're an employee of the community centre, but it seems obvious that they're a volunteer doing volunteer work at the community centre - and NY law says that either the community centre is a non-profit, or it can't have volunteers; if that's a for-profit organization then that guy running a debate club must be paid minimum wage.
I definitely shouldn't be subject to labor laws for that work. It's organizing stuff around a hobby, not creating concrete value for or at the discretion of Reddit the business.
Some of the bigger subs though should have full time staff on the mod teams however.
I believe the key difference also is that you can just not do your work without reddit really minding (that's how some subs have died). That's very different from Stack Exchange.